Mathematics Education Research Journal

, Volume 28, Issue 3, pp 379–404 | Cite as

Reasoning about variables in 11 to 18 year olds: informal, schooled and formal expression in learning about functions

  • Michal AyalonEmail author
  • Anne Watson
  • Steve Lerman
Original Article


This study examines expressions of reasoning by some higher achieving 11 to 18 year-old English students responding to a survey consisting of function tasks developed in collaboration with their teachers. We report on 70 students, 10 from each of English years 7–13. Iterative and comparative analysis identified capabilities and difficulties of students and suggested conjectures concerning links between the affordances of the tasks, the curriculum, and students’ responses. The paper focuses on five of the survey tasks and highlights connections between informal and formal expressions of reasoning about variables in learning. We introduce the notion of ‘schooled’ expressions of reasoning, neither formal nor informal, to emphasise the role of the formatting tools introduced in school that shape future understanding and reasoning.


Reasoning about variables Functions Informal Schooled Formal 


  1. Ainley, J., & Pratt, D. (2005). The Significance of task design in mathematics education: examples from proportional reasoning. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29 International Conference for the Psychology of Mathematics Education (1st ed., pp. 93–122).Google Scholar
  2. Ayalon, M., Watson, A., & Lerman, S. (2015). Progression towards functions: identifying variables and relations between them. International Journal of Science and Mathematics Education (published online).Google Scholar
  3. Ayalon, M., Watson, A., & Lerman, S. (2016). Students’ conceptualisations of function revealed through definitions and examples. In press.Google Scholar
  4. Berger, M. (2005). Vygotsky’s theory of concept formation and mathematics education (2nd ed.). Bergen, Norway: Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education.Google Scholar
  5. Blanton, M., & Kaput, J. (2005). Helping elementary teachers build mathematical generality into curriculum and instruction. ZDM – The International Journal on Mathematics Education, 37(1), 34–42.Google Scholar
  6. Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: a framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378.CrossRefGoogle Scholar
  7. Carlson, M., & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Research Sampler Series, 9, The Mathematical Association of America Notes Online. Retrieved from
  8. Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment: a tool for assessing students’ reasoning abilities and understandings. Cognition and Instruction, 28(2), 113–145.CrossRefGoogle Scholar
  9. Clement, J. (1985). Misconceptions in graphing. Proceedings of the 9th Conference of the International Group for the Psychology of Mathematics Education, 1, 369–375.Google Scholar
  10. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26, 135–164.CrossRefGoogle Scholar
  11. Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86.CrossRefGoogle Scholar
  12. Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovi’c, D. (1996). Understanding the limit concept: beginning with a coordinated process schema. Journal of Mathematical Behavior, 15, 167–192.CrossRefGoogle Scholar
  13. Dreyfus, T., & Eisenberg, T. (1983). The function concept in college students: linearity, smoothness and periodicity. Focus on Learning Problems in Mathematics, 5, 119–132.Google Scholar
  14. Dowling, P. C., & Brown, A. J. (2010). Doing research/reading research: re-interrogating education. London: Routledge.Google Scholar
  15. Goldenberg, E. P. (1987). Believing is seeing: How preconceptions influence the perceptions of graphs. In J. Bergeron, N. Herscovits, & C. Kieran (Eds.), Proceedings of the Eleventh Conference of the International group for the Psychology of Mathematics Education (1st ed., pp. 197–203).Google Scholar
  16. Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: a calculus course as an example. Educational Studies in Mathematics, 39, 111–129.CrossRefGoogle Scholar
  17. Herbert, S., & Pierce, P. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics, 81, 85–101.CrossRefGoogle Scholar
  18. Janvier, C. (1981). Use of situations in mathematics education. Educational Studies in Mathematics, 12, 113–122.CrossRefGoogle Scholar
  19. Kaput, J. J. (1992). Patterns in students’ formalization of quantitative patterns. In G. Harel & E. Dubinsky (Eds.), The concept of function: aspects of epistemology and pedagogy (pp. 290–318). Washington, DC: Mathematical Association of America.Google Scholar
  20. Karplus, R. (1978). Intellectual development beyond elementary school IX: Functionality, a survey (Advancing Education through Science Oriented Programs, Report ID-51). Berkeley: University of California at Berkeley.Google Scholar
  21. Kuntze, S., Lerman, S., Murphy, B., Kurz-Milcke, E., Siller, H.-S., & Winbourne, P. (2011). Development of preservice teachers’ knowledge related to big ideas in mathematics. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 105–112). Ankara, Turkey: PME.Google Scholar
  22. Leinhardt, G., Zaslavsky, O., & Stein, M. (1990). Functions, graphs and graphing: tasks, learning and teaching. Review of Educational Research, 60(1), 37–42.CrossRefGoogle Scholar
  23. Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA Notes, 25 (pp. 175–193). Washington, DC: Mathematical Association of America.Google Scholar
  24. Monk, S., & Nemirovsky, R. (1994). The case of Dan: student construction of a functional situation through visual attributes. CBMS Issues in Mathematics Education, 4, 139–168.CrossRefGoogle Scholar
  25. Nemirovsky, R. (1996). A functional approach to algebra: two issues that emerge. In N. Dedrarg, C. Kieran, & L. Lee (Eds.), Approaches to algebra: perspectives for research and teaching (pp. 295–313). Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
  26. Newman, F., & Holzman, L. (1993). Lev Vygotsky: revolutionary scientist. London: Routledge.Google Scholar
  27. Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: a semiotic analysis. Educational Studies in Mathematics, 42, 237–268.CrossRefGoogle Scholar
  28. Slavit, D. (1997). An alternative route to reification of a function. Educational Studies in Mathematics, 33, 259–281.CrossRefGoogle Scholar
  29. Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164.CrossRefGoogle Scholar
  30. Steele, D. (2008). Seventh-grade students’ representations for pictorial growth and change problems. ZDM– International Journal in Mathematics Education, 40, 97–110.CrossRefGoogle Scholar
  31. Swan, M. (1980). The language of functions and graphs. Nottingham, UK: Shell Centre for Mathematical Education. University of Nottingham.Google Scholar
  32. Thompson, P. W. (1994a). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229–274.Google Scholar
  33. Thompson, P. W. (1931b). Students, functions, and the undergraduate mathematics curriculum. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in collegiate mathematics education (Vol. 4, pp. 21–44). Providence, RI: American Mathematical Society.Google Scholar
  34. Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23, 57–86.CrossRefGoogle Scholar
  35. Vygotsky, L. S. (1994/1931). The development of thinking and concept formation in adolescence. In R. Van der Veer & J. Valsiner (Eds.), The Vygotsky reader (pp. 185–265). Oxford: Blackwell Publishers.Google Scholar
  36. Vygotsky, L. S. (1978). Mind in Society. London: Harvard University Press.Google Scholar
  37. Warren, E., & Cooper, T. (2007). Generalising the pattern rule for visual growth patterns: actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67(2), 171–185.CrossRefGoogle Scholar
  38. Wilmot, D. B., Schoenfeld, A. H., Wilson, M., Champney, D., & Zahner, W. (2011). Validating a learning progression in mathematical functions for college readiness. Mathematical Thinking and Learning, 13(4), 259–291.CrossRefGoogle Scholar
  39. Zandieh, M. (2000). A theoretical framework for analyzing student understanding the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in collegiate mathematics education, IV (Vol. 8, pp. 103–127). Providence, RI: American Mathematical Society.CrossRefGoogle Scholar

Copyright information

© Mathematics Education Research Group of Australasia, Inc. 2016

Authors and Affiliations

  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.University of OxfordOxfordUK
  3. 3.London South Bank UniversityLondonUK

Personalised recommendations